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>But the part in brackets is the same as the whole, and hence is equal to 2. Thus we have 2=x^2 The bug is in the third line, the statement is invalid. Obviou

>But the part in brackets is the same as the whole, and hence is equal to 2. Thus we have 2=x^2 The bug is in the third line, the statement is invalid. Obviously (x^x infinitum) is not the same as (x^x infinitum-1). For the same reason that x^x^x^x != x^x^x. Sadly, I'm not math savvy. In fact the highest math level I passed in college was calc 3 and that was after taking calc 1 and calc 2 twice. So, my answer is based on logical reasoning rather than mathematical proof. Spock would likely indicate my logic is flawed, and only co-incidentally leads to the correct conclusion. Either way, this doesn't seem like a very interesting "math paradox".

RiderOfGiraffes 16y ago
I'm really pleased that you've been willing to stick your head above the parapet and say this. What you've said seems to me to be a common line of reasoning, but it always confuses me. In particular, you say: > Sadly, I'm not math savvy. OK, that's cool, and I'm always willing to engage with people who want to learn. > Obviously (x^x infinitum) is not the same as (x^x infinitum-1). Given that you're not math savvy, that's a pretty definite thing to say. > ... my answer is based on logical reasoning rather than mathematical proof. Well, actually your answer seems to be based on intuition, and not on reasoning at all. You intuitively say that x^x^x^x != x^x^x, which is fair enough, and then leap to the conclusion that the infinite case must be like the finite case. And there's a problem. The infinite case is not like the finite case. Similarly, the infinite sum 1/2+1/4+1/8+1/16+1/32+... insofar as it can be given any meaning at all, must be set equal to 1, even though all the finite partial sums are strictly less than 1. > this doesn't seem like a very interesting "math paradox". But your "reasoning" is wrong. Perhaps you've dismissed it as "uninteresting" because you haven't really understood it? That's fair enough - I don't have a problem with that, but it would be nice to believe that you realised it. Let me just say again that I'm really pleased you expressed your thoughts on this. I'm sure you're not the only one, and I find it useful to hear the way people think about these things. I give lots of presentations and masterclasses on math to all ranges of ages and experience. I need to know the intuitions people are using. So thanks.
- giberson 16y ago
  Thanks for your response Rider, and let me re-iterate that I'm already aware my reasoning is wrong. Hence my allusion to how I think Spock would reply to me. For clarification, I'm considering my observation that x^x infinitum is not the same as x^x infinitum-1 as based on reason rather than intuition because there is reasoning behind my statement. This may be an argument of semantic and thats certainly what I want to avoid since it's useless. But my answer isn't based on an instinctive knowing (intuition) it was based on the following considerations: 3^1 = 3 3^2 = 9 3^3 = 27 For any n^x I could* think of, n^(x-1) was a different value. If I think of infinity as some unbounded large number, then I think of x-1 as a smaller (by 1) unbounded large number. It may be fair to consider this an intuition, but since I'm basing it off of observations of smaller number examples I think its closer to an implication of the previous observed behavior. With that argument I contest that my conclusion is in fact based on reason. However, I will and still do submit that my reasoning is that of a layman and may be (and probably is) incorrect. But, my conclusion seems to be inline with that of the other comments to this thread regarding the overall evaluation 2=4 through this number game. Now, I'm making an assumption here and by all rights perhaps making an ass of myself, but from my understanding of the majority responses is an agreement that 2=x^x(infinitum) does not lead to a conclusion of 2 != 4. Their arguments are more technical and perhaps outside my understanding level, but I think this is where my point lies about the math paradox being interesting. I would think that the paradox would only be interesting if the mathematical reasoning would be counter intuitive than a layman's intuition. That is to say, if the layman says it should be yes, mathematically it should be no in order for it to be interesting. (Again, I'm making the assumption original post argument doesn't hold water based on the refutation of the comment thread) Thanks for taking the time to reply, level headed even without being insulting, to my [mathematically incompetent] comments. :) * while formulating this reply, it occurred to me that if n = 1, 1^x = 1^(x-1). Here during this argument I've discovered a flaw in my reasoning. Which I am conceding in my foot note as my comment above is meant to explain my original reasoning.
jerf 16y ago
This isn't the only or necessarily the best way to think of infinity, but I think it's one of the better introductions to the idea: Think of infinity not as a concrete number (which it isn't, but without a replacement concept it's very easy to treat it as one because you know nothing else), but a statement that the answer to the question "Is there another?" is always "Yes". If you have 3, you can ask "Is there another?" and get three yeses, then you get no more. For 5, you get 5 yeses, then no more. For infinity, you never get told no. There's no difference between infinity and infinity minus 1, because even after asking once "Is there another?" of your infinity, you've still got something that will forever answer "Yes" just as you did before. This is also useful in things like proofs, because of that property. For any given finite instance of whatever thing you're trying to prove, you will never fully "use up" the infinity, but because of the way infinity never says "no", it allows you to write the proof without having to handle the cases where you run out of something, which can make things a lot trickier. Similarly, "taking the limit as x goes to infinity" is said that way because x never reaches infinity, it's never "equal to" infinity, but for any number you choose, you can ask the question "Is x there yet?" and the answer is "No", always.
- giberson 16y ago
  I contest that infinity != infinity-1. I base this on an assumption that there is more than one "infinity". Additionally, I will contest infinity != infinity necessarily. If we count on the number line from 1 to X, never stopping, that would be an infinite process. If we count on the number line from 1 to 2 by using fractional numbers that infinitely decrease, that is also an infinite process. However, both these infinite outcomes involve different components (different number values) and as such I consider them "different" infinities. I admit this is a fairly week argument, and much closer to "intuition" than my original comment. But, can you provide a mathematical proof that infinity = infinity-1?
  - jerf 16y ago
    One infinity is not necessarily equal to another. You are correct. Equality is not a very useful concept for infinities, really, you need other concepts like one-to-one mappings. Take one away from your infinity and you haven't done anything to its cardinality. A one-to-one mapping between the old infinity and the new one is still trivially available. Words are useless. English is absolutely incapable of dealing with infinities in math, full stop. If you are arguing with English words, you're already wrong. That is why I said that I am presenting one useful way to think of them, and it is absolutely not the best or only way. What I showed was just a somewhat intuitive way to think of the infinity most commonly encountered in otherwise-conventional math and isn't totally correct there, either, it's just intuitively closer. Arguing about my intuitive shortcut misses the point entirely. You must go learn the math. I can't use the math terms to explain it to you here, either, because the correct understanding of the terminology would be tautological to understanding my point. Also, I am not trying to be mean here. It took mathematicians decades to work out consistent definitions of infinity, and if you are going to insist on banging on the definitions and exposing the inconsistencies, an admirable enterprise, the only definitions that will stand up to that scrutiny are the ones that have been honed over the decades, and they do not fit well into HN posts. (Even if they are posted, they are thoroughly based on other mathematics that they would be meaningless until you understand those other things.) There's no Royal Road to infinity.

RiderOfGiraffes 16y ago

I'm really pleased that you've been willing to stick your head above the parapet and say this. What you've said seems to me to be a common line of reasoning, but it always confuses me. In particular, you say: > Sadly, I'm not math savvy. OK, that's cool, and I'm always willing to engage with people who want to learn. > Obviously (x^x infinitum) is not the same as (x^x infinitum-1). Given that you're not math savvy, that's a pretty definite thing to say. > ... my answer is based on logical reasoning rather than mathematical proof. Well, actually your answer seems to be based on intuition, and not on reasoning at all. You intuitively say that x^x^x^x != x^x^x, which is fair enough, and then leap to the conclusion that the infinite case must be like the finite case. And there's a problem. The infinite case is not like the finite case. Similarly, the infinite sum 1/2+1/4+1/8+1/16+1/32+... insofar as it can be given any meaning at all, must be set equal to 1, even though all the finite partial sums are strictly less than 1. > this doesn't seem like a very interesting "math paradox". But your "reasoning" is wrong. Perhaps you've dismissed it as "uninteresting" because you haven't really understood it? That's fair enough - I don't have a problem with that, but it would be nice to believe that you realised it. Let me just say again that I'm really pleased you expressed your thoughts on this. I'm sure you're not the only one, and I find it useful to hear the way people think about these things. I give lots of presentations and masterclasses on math to all ranges of ages and experience. I need to know the intuitions people are using. So thanks.

giberson 16y ago
Thanks for your response Rider, and let me re-iterate that I'm already aware my reasoning is wrong. Hence my allusion to how I think Spock would reply to me. For clarification, I'm considering my observation that x^x infinitum is not the same as x^x infinitum-1 as based on reason rather than intuition because there is reasoning behind my statement. This may be an argument of semantic and thats certainly what I want to avoid since it's useless. But my answer isn't based on an instinctive knowing (intuition) it was based on the following considerations: 3^1 = 3 3^2 = 9 3^3 = 27 For any n^x I could* think of, n^(x-1) was a different value. If I think of infinity as some unbounded large number, then I think of x-1 as a smaller (by 1) unbounded large number. It may be fair to consider this an intuition, but since I'm basing it off of observations of smaller number examples I think its closer to an implication of the previous observed behavior. With that argument I contest that my conclusion is in fact based on reason. However, I will and still do submit that my reasoning is that of a layman and may be (and probably is) incorrect. But, my conclusion seems to be inline with that of the other comments to this thread regarding the overall evaluation 2=4 through this number game. Now, I'm making an assumption here and by all rights perhaps making an ass of myself, but from my understanding of the majority responses is an agreement that 2=x^x(infinitum) does not lead to a conclusion of 2 != 4. Their arguments are more technical and perhaps outside my understanding level, but I think this is where my point lies about the math paradox being interesting. I would think that the paradox would only be interesting if the mathematical reasoning would be counter intuitive than a layman's intuition. That is to say, if the layman says it should be yes, mathematically it should be no in order for it to be interesting. (Again, I'm making the assumption original post argument doesn't hold water based on the refutation of the comment thread) Thanks for taking the time to reply, level headed even without being insulting, to my [mathematically incompetent] comments. :) * while formulating this reply, it occurred to me that if n = 1, 1^x = 1^(x-1). Here during this argument I've discovered a flaw in my reasoning. Which I am conceding in my foot note as my comment above is meant to explain my original reasoning.

jerf 16y ago

This isn't the only or necessarily the best way to think of infinity, but I think it's one of the better introductions to the idea: Think of infinity not as a concrete number (which it isn't, but without a replacement concept it's very easy to treat it as one because you know nothing else), but a statement that the answer to the question "Is there another?" is always "Yes". If you have 3, you can ask "Is there another?" and get three yeses, then you get no more. For 5, you get 5 yeses, then no more. For infinity, you never get told no. There's no difference between infinity and infinity minus 1, because even after asking once "Is there another?" of your infinity, you've still got something that will forever answer "Yes" just as you did before. This is also useful in things like proofs, because of that property. For any given finite instance of whatever thing you're trying to prove, you will never fully "use up" the infinity, but because of the way infinity never says "no", it allows you to write the proof without having to handle the cases where you run out of something, which can make things a lot trickier. Similarly, "taking the limit as x goes to infinity" is said that way because x never reaches infinity, it's never "equal to" infinity, but for any number you choose, you can ask the question "Is x there yet?" and the answer is "No", always.

giberson 16y ago
I contest that infinity != infinity-1. I base this on an assumption that there is more than one "infinity". Additionally, I will contest infinity != infinity necessarily. If we count on the number line from 1 to X, never stopping, that would be an infinite process. If we count on the number line from 1 to 2 by using fractional numbers that infinitely decrease, that is also an infinite process. However, both these infinite outcomes involve different components (different number values) and as such I consider them "different" infinities. I admit this is a fairly week argument, and much closer to "intuition" than my original comment. But, can you provide a mathematical proof that infinity = infinity-1?
- jerf 16y ago
  One infinity is not necessarily equal to another. You are correct. Equality is not a very useful concept for infinities, really, you need other concepts like one-to-one mappings. Take one away from your infinity and you haven't done anything to its cardinality. A one-to-one mapping between the old infinity and the new one is still trivially available. Words are useless. English is absolutely incapable of dealing with infinities in math, full stop. If you are arguing with English words, you're already wrong. That is why I said that I am presenting one useful way to think of them, and it is absolutely not the best or only way. What I showed was just a somewhat intuitive way to think of the infinity most commonly encountered in otherwise-conventional math and isn't totally correct there, either, it's just intuitively closer. Arguing about my intuitive shortcut misses the point entirely. You must go learn the math. I can't use the math terms to explain it to you here, either, because the correct understanding of the terminology would be tautological to understanding my point. Also, I am not trying to be mean here. It took mathematicians decades to work out consistent definitions of infinity, and if you are going to insist on banging on the definitions and exposing the inconsistencies, an admirable enterprise, the only definitions that will stand up to that scrutiny are the ones that have been honed over the decades, and they do not fit well into HN posts. (Even if they are posted, they are thoroughly based on other mathematics that they would be meaningless until you understand those other things.) There's no Royal Road to infinity.